Abstract
For the problem of estimating the normal variance σ2 based on random sampleX 1,...,X n when a preliminary conjectured interval [C −10 σ 20 ,C 0σ 20 ] is available, the minimum discrimination information (MDI) approach is presented. This provides a simple way of specifying the prior information, and also allows to consider a shrinkage type estimator. MDI estimator and its mean square error are derived. The estimator compares favorably with the previously proposed estimators in terms of mean square error efficiency.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Bancroft, T. A. (1944). On biases in estimation due to the use of preliminary tests of significance,Ann. Math. Statist.,15, 190–204.
Bickel, P. J. (1981). Minimax estimation of the mean of a normal distribution when the parameter space is restricted,Ann. Statist.,9, 1301–1309.
Casella, G. and Strawderman, W. E. (1981). Estimating a bounded mean,Ann. Statist.,9, 870–878.
Escobar, L. A. and Skarpness, B. (1987). Mean square error and efficiency of the least squares estimator over interval constraints,Comm. Statist. Theory Methods,16, 397–406.
Gelfand, A. E. and Dey, D. K. (1988). Improved estimation of the disturbance variance in a linear regression model,J. Econometrics,39, 387–395.
Gokhale, D. V. and Kullback, S. (1978). The minimum discrimination information approach in analyzing categorical data,Comm. Statist. Theory Methods,7, 987–1005.
Hirano, K. (1978). A note on the level of significance of the preliminary test in pooling variances,J. Japan Statist. Soc.,8, 71–75.
Hirano, K. (1980). On the estimators for estimating variance of a normal distribution,Recent Developments in Statistical Inference and Data Analysis (ed. K. Matusita), 105–115, North-Holland, Amsterdam.
Inada, K. (1989). Estimation of variance after preliminary conjecture,Bull. Inform. Cybernet.,23, 183–198.
Klemm, R. J. and Sposito, V. A. (1980). Least squares solutions over interval restrictions,Comm. Statist. Simulation Comput.,9, 423–425.
Kullback, S. and Leibler, R. A. (1951). On information and sufficiency,Ann. Math. Statist.,22, 79–86.
Mendenhall, W. (1979).Introduction to Probability and Statistics, 5th ed., Ouxbury Press, Bdmont.
Ohtani, K. (1991). Estimation of the variance in a normal population after the one-sided pre-test for the mean,Comm. Statist. Theory Methods,20, 219–234.
Ohtani, K. and Toyoda, T. (1978). Minimax regret critical values for a preliminary test in pooling variances,J. Japan Statist. Soc.,8, 15–20.
Pandey, B. N. and Mishra, G. C. (1991). Weighted estimators for a normal population variance,Comm. Statist. Theory Methods,20, 235–247.
Shore, J. E. and Johnson, R. W. (1980). Axiomatic derivation of the principle of maximum entropy and the principle of minimum cross-entropy,IEEE Trans. Inform. Theory,26, 26–37.
Toyoda, T. and Wallace, T. D. (1975). Estimation of variance after a preliminary test of homogeneity and optimal levels of significance for the pre-test,J. Econometrics,3, 395–404.
Author information
Authors and Affiliations
About this article
Cite this article
Gokhale, D.V., Inada, K. & Kim, HJ. A minimum discrimination information estimator of preliminary conjectured normal variance. Ann Inst Stat Math 45, 129–136 (1993). https://doi.org/10.1007/BF00773673
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF00773673